On H-Supermagic Labelings of m-Shadow of Paths and Cycles

Abstract

A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,…,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,…,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,…,Gm, and then joining every vertex u in Gi, i∈{1,2,…,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles.